Modeling and characterizing anisotropic inclusion orientation in heterogeneousmaterial via directional cluster functions and stochastic microstructure reconstructionYang Jiao and Nikhilesh Chawla Citation: Journal of Applied Physics 115, 093511 (2014); doi: 10.1063/1.4867611 View online: http://dx.doi.org/10.1063/1.4867611 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characterizing randomly anisotropic surfaces in eddy-current NDE AIP Conf. Proc. 1430, 340 (2012); 10.1063/1.4716248 RECENT ADVANCES IN MODELING DISCONTINUITIES IN ANISOTROPIC AND HETEROGENEOUSMATERIALS IN EDDY CURRENT NDE AIP Conf. Proc. 1335, 1565 (2011); 10.1063/1.3592116 Surface and microstructural characterization of laser beam welds in an aluminum alloy J. Vac. Sci. Technol. A 20, 1416 (2002); 10.1116/1.1487868 Role of anisotropy in noncontacting thermoelectric materials characterization J. Appl. Phys. 91, 225 (2002); 10.1063/1.1416852 Anisotropic effective conductivity of materials with nonrandomly oriented inclusions of diverse ellipsoidal shapes J. Appl. Phys. 87, 8561 (2000); 10.1063/1.373579

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Modeling and characterizing anisotropic inclusion orientation inheterogeneous material via directional cluster functions and stochasticmicrostructure reconstruction

Yang Jiaoa) and Nikhilesh ChawlaMaterials Science and Engineering, Arizona State University, Arizona 85287-6206, USA

(Received 31 October 2013; accepted 22 February 2014; published online 6 March 2014)

We present a framework to model and characterize the microstructure of heterogeneous materials

with anisotropic inclusions of secondary phases based on the directional correlation functions of

the inclusions. Specifically, we have devised an efficient method to incorporate both directional

two-point correlation functions S2 and directional two-point cluster functions C2 that contain

non-trivial topological connectedness information into the simulated annealing microstructure

reconstruction procedure. Our framework is applied to model an anisotropic aluminum alloy and

the accuracy of the reconstructed structural models is assessed by quantitative comparison with the

actual microstructure obtained via x-ray tomography. We show that incorporation of directional

clustering information via C2 significantly improves the accuracy of the reconstruction. In addition,

a set of analytical “basis” correlation functions are introduced to approximate the actual S2 and C2

of the material. With the proper choice of basis functions, the anisotropic microstructure can be

represented by a handful of parameters including the effective linear sizes of the iron-rich and

silicon-rich inclusions along three orthogonal directions. This provides a general and efficient

means for heterogeneous material modeling that enables one to significantly reduce the data set

required to characterize the anisotropic microstructure. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4867611]

I. INTRODUCTION

An accurate knowledge of the microstructure of a

heterogeneous material is crucial for subsequent modeling

and prediction of its physical properties and performance.

In the past few decades, a quantitative understanding of

the structure-property relation in such heterogeneous materi-

als has begun to emerge, mainly due to the development of

advanced experimental and computational material micro-

structure characterization techniques.1–3 In particular, x-ray

tomography technique,4–6 which eliminates destructive

cross-sectioning and allows for superior resolution and

image quality with minimal sample preparation,7,8 has been

widely used to obtain high-resolution three-dimensional

(3D) microstructure for a wide range of heterogeneous mate-

rials, including Sn-rich alloys,9 powder metallurgy steels,10

metal matrix composites,11–15 and lightweight alloys.16–20

The resulting microstructural data sets can be subsequently

incorporated into finite element models to predict the onset

of local damage mechanisms and the macroscopic deforma-

tion behavior.21–25

Experimentally obtained microstructural data are

usually represented as a large array whose entries indicate

the local states of the microstructure (e.g., the phase that a

specific voxel belongs to). Although morphological details

of a specific material can be contained in its microstructural

array, such information would not all be useful for macro-

scopic effective property analysis, which requires averaging

over a sufficiently large number of material samples to yield

meaningful and robust statistics. The structural details asso-

ciated with a specific material sample usually do not signifi-

cantly contribute to the overall statistics. In addition, in the

case where the entire course of microstructural evolution is

of interest (e.g., during the solidification or coarsening

processes), the resulting 4D structural data sets (three spatial

dimensions plus one temporal dimension) are usually

extremely large.

Therefore, it is highly desirable to devise a robust frame-

work that enables one to represent, characterize, and model

the complex microstructure of a heterogeneous material in a

much more efficient manner. Recently, it has been suggested

that certain statistical morphological descriptors, such as

lower-order spatial correlation functions of the material’s

phases, can be employed to model a wide spectrum of com-

plex microstructures.26–30 This allows one to reduce the large

data sets for a complete specification of all of the local states

in a microstructure to a handful of simple scalar functions

that statistically capture the salient structural features.

Stochastic reconstruction techniques such as the simulated

annealing procedure developed by Yeong and Torquato31,32

can then be employed to generate realistic virtual 3D micro-

structures from the correlation functions. Material modeling

and reconstruction based on correlation functions has

enabled the development of data-driven material design

schemes33 and structure-based performance modeling.34

In this paper, we present a framework to model hetero-

geneous materials with anisotropic inclusions of secondary

phases based on the directional correlation functions of the

inclusions. In particular, we devise an efficient method that

a)Author to whom correspondence should be addressed. Electronic mail:

yang.jiao.2@asu.edu

0021-8979/2014/115(9)/093511/9/$30.00 VC 2014 AIP Publishing LLC115, 093511-1

JOURNAL OF APPLIED PHYSICS 115, 093511 (2014)

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enables us to incorporate directional two-point cluster func-

tions C2 into the reconstruction procedure, which contain

non-trivial topological connectedness information of the

material’s phases.28,35 We apply our general method to

model the microstructure of an Al alloy with anisotropic

inclusions of Fe-rich and Si-rich phases. We show quantita-

tively that incorporating such clustering information can lead

to an improved reconstruction by comparing the lineal-path

functions of the actual and reconstructed microstructures. In

addition, we introduce a set of analytical “basis” correlation

functions to approximate the actual S2 and C2 of the material.

This enables us to further reduce the set of parameters

required to characterize the anisotropic material microstruc-

ture. Specifically, we show that with our choice of basis

functions, a heterogeneous microstructure can be represented

by a handful of parameters including the effective linear

sizes of the Fe-rich and Si-rich inclusions along three orthog-

onal directions.

The rest of the paper is organized as follows: In

Sec. II, we provide definitions of the statistical morpho-

logical descriptors (i.e., correlation functions) employed

to model the alloy microstructure. In Sec. III, we present

the Yeong-Torquato (YT) reconstruction procedure and

our method for efficient sampling of directional correla-

tion functions from trial microstructures. In Sec. IV, we

employ the functions S2 and C2 and the YT reconstruction

procedure to model an anisotropic alloy microstructure.

In addition, we introduce a set of analytical “basis” corre-

lation functions associated with elongated particles to

approximate the actual S2 and C2 of the material to fur-

ther reduce the set of parameters required to characterize

the heterogeneous microstructure. Concluding remarks

are given in Sec. V.

II. STATISTICAL MICROSTRUCTURAL DESCRIPTORS

A. Two-point correlation function

In general, the microstructure of an Al alloy with aniso-

tropic secondary phase inclusions in the Al matrix can be

uniquely determined by specifying the indicator functions

associated with all of the individual phases (the inclusions

and the matrix),1 i.e.,

I ið ÞðxÞ ¼1 x in phase i

0 otherwise;

((1)

where i¼ 1,…, q and q is the total number of phases. The

volume fraction of phase i is then given by

ui ¼ hI ið ÞðxÞi; (2)

where hi denotes the ensemble average over many independ-

ent material samples or volume average over a single large

sample if it is spatially “ergodic.”1 The two-point correlation

function SðijÞ2 ðx1; x2Þ associated with phases i and j is defined

as

SðijÞ2 ðx1; x2Þ ¼ hI ið Þðx1ÞI jð Þðx2Þi (3)

which also gives the probability that two randomly selected

points x1 and x2 fall into phases i and j, respectively

(Fig. 1(a)). For a material with q distinct phases, there are

totally q� q different S2. However, it has been shown that

only q of them are independent1,36 and the remaining q� (q� 1) functions can be explicitly expressed in terms of

the q independent ones. In our case, q¼ 3 (Fe-rich inclu-

sions, Si-rich inclusions, and the Al matrix) and thus, we

only need to consider the two-point correlation functions

associated with the anisotropic inclusions, including the

auto-correlation function of the Fe-rich inclusions SFe2 , the

auto-correlation function of the Si-rich inclusions SSi2 , and

cross correlation function SFe�Si2 .

For statistically homogeneous materials such as the Al

alloy considered here, there is no preferred center in the

microstructure. Therefore, the associated S2 depends only on

the relative vector displacement between the two points,1

i.e.,

S2ðx1; x2Þ ¼ S2ðx2 � x1Þ ¼ S2ðrÞ; (4)

where r¼ x2� x1. At r¼ 0, the auto-correlation function

gives the probability that a randomly selected point falls into

the phase of interest, i.e., the volume fraction of the associ-

ated phase. However, the cross correlation function is zero at

r¼ 0, since the probability of finding a single point falling

into two different phases is zero. At large r values, the

FIG. 1. Schematic illustration of the events that contribute to various correlation functions. The two-point correlation function S2 gives the probably of finding

two points in the phases of interest. In (A), we show events that contribute to S2(ij), where i, j can be either red phase or blue phase. The lineal-path function

L(r) gives the probability that a randomly chosen line segment of length r entirely falls into the phase of interest. In (B), we show events that contribute to L(i).

The two-point cluster function C2(r) gives the probably of finding two points separated by r in the same cluster of the phase of interest. In (C), we show events

that contribute to C2(i).

093511-2 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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probabilities of finding the two points in the phases of inter-

est are independent of one another, leading to u2i for the

auto-correlation functions and uiuj for the cross correlation

function.

For a statistically isotropic material, S2 depends only on

the scalar distance between a pair of points. In the case of

the Al alloy, the secondary phase inclusions are significantly

elongated along the rolling direction (RD), leading to an

overall anisotropic microstructure with a preferred direction.

Therefore, we employ the S2 along three orthogonal direc-

tions, i.e., the RD, transverse (TD), and normal (ND) direc-

tions, to characterize the microstructure. Accordingly, we

denote the directional two-point correlation function of type

a along direction b by Sa;b2 ðrÞ, where a¼ {Fe, Si, Fe-Si} and

b¼ {RD, TD, ND}. We note that the general n-point correla-

tion function Sn which gives the probability of finding a

particular n-point configuration in specific phases can be

defined in a similar manner as S2; see Eq. (2). It has been

shown that the effective properties of a heterogeneous mate-

rial can be explicitly expressed as series expansions involv-

ing certain integrals of Sn. Interested readers are referred to

Ref. 2 for detailed discussions of Sn and their properties.

B. Lineal-path function

The lineal-path function L(i)(r) gives probability that a

randomly selected line segment of length r¼ |r| along the

direction of vector r entirely falls into phase i (Fig. 1(b)).1,37

At r¼ 0, L(i)(0) reduces to the probability of finding a point

in phase i and thus, L(i)(0)¼ui. In materials that do not con-

tain system-spanning clusters, the chance of finding a line

segment with very large length entirely falling into any

phases is vanishingly small. Accordingly, for large r values,

L(i) decays to zero rapidly in such materials. The lineal-path

function contains partial topological connectedness informa-

tion of the material’s phases, i.e., that along a lineal-path.

Generally, the lineal-path function underestimates the degree

of clustering in the system (e.g., two points belonging to the

same cluster but not along a specific lineal path will not con-

tribute to L). For the anisotropic alloy microstructure consid-

ered here, we use the directional lineal-path functions La;bðrÞfor the Fe-rich and Si-rich inclusions along three orthogonal

directions, where a¼ {Fe, Si} and b¼ {RD, TD, ND}.

C. Two-point cluster function

The two-point cluster correlation function CðiÞ2 ðx1; x2Þ

gives the probability that two randomly selected points x1 and

x2 fall into the same cluster of phase i (Fig. 1(c)).1,35 A cluster

of a phase is defined as a compact region in which any point

in the region is connected to any other point in the region by a

continuous path completely in the region. For statistically ho-

mogeneous materials, C2 depends only on the relative vector

displacement between the two points, i.e., C2ðx1; x2Þ ¼ C2ðrÞ.In contrast to the lineal-path function, C2 contains complete

clustering information of the phases, which has been shown to

have dramatic effects on the material’s physical properties.1

Moreover, unlike S2 and L, the cluster functions generally can-

not be obtained from lower-dimensional cuts (e.g., 2D slices)

of a 3D microstructure, which may not contain correct con-

nectedness information of the actual 3D system.

It has been shown that C2 is related to S2 via the follow-

ing equation:35

SðiiÞ2 ðrÞ ¼ C

ðiÞ2 ðrÞ þ D

ðiÞ2 ðrÞ; (5)

where DðiÞ2 ðrÞ measures the probability that two points sepa-

rated by r fall into different clusters of the phase of interest.

In other words, C2 is the connectedness contribution to the

standard two-point correlation function S2. As in the case of

S2 and L, we employ directional cluster functions Ca;b2 ðrÞ to

characterize the anisotropic alloy, where a¼ {Fe, Si} and

b¼ {RD, TD, ND}. For microstructures with well-defined

inclusion, C2(r) of the inclusions is a short-ranged function

that rapidly decays to zero as r approaches the largest linear

size of the inclusions. We note that although C2 is a

“two-point” quantity, it has been shown to embody

higher-order structural information which makes it a highly

sensitive statistical descriptor over and above S2.28,38

III. STOCHASTIC MICROSTRUCTURERECONSTRUCTION PROCEDURE

A. Simulated annealing procedure

We use the YT reconstruction procedure31,32 to generate

virtual 3D microstructures from a specific set of directional

correlation functions discussed in the previous section. We

note that there are many other different microstructure recon-

struction procedures, such as the Gaussian random field

method,39 phase recovery method,40 and the recently devel-

oped raster path method.41 However, the YT procedure ena-

bles one to incorporate an arbitrary number of correlations of

any type, while the others require specific structural informa-

tion as input.

In the YT procedure, the reconstruction problem is

formulated as an “energy” minimization problem, with the

energy functional E defined as follows:

E ¼X

a

Xb

Xr

f a;bðrÞ � �fa;bðrÞ

h i2

; (6)

where �fa;bðrÞ is a target correlation function of type a along

direction b and f a;bðrÞ is the corresponding function associ-

ated with a trial microstructure. The simulated annealing

method42 is usually employed to solve the aforementioned

minimization problem. Specifically, starting from an initial

trial microstructure (i.e., old microstructure) which contains

a fixed number of voxels for each phase consistent with the

volume fraction of that phase, two randomly selected voxels

associated with different phases are exchanged to generate a

new trial microstructure. Relevant correlation functions are

sampled from the new trial microstructure and the associated

energy is evaluated, which determines whether the new trial

microstructure should be accepted or not via the probability

paccðold!newÞ¼min 1; expEold

T

� ��exp

Enew

T

� �( ); (7)

093511-3 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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where T is a virtual temperature that is chosen to be high ini-

tially and slowly decreases according to a cooling sched-

ule.31,32 The above process is repeated until E is smaller than

a prescribed tolerance, which we choose to be 10�10 here.

Generally, several hundred thousand trials need to be made

to achieve such a small tolerance.

B. Efficient sampling of correlation functions fromtrial microstructure

It can be clearly seen that the key step in the reconstruc-

tion process is to efficiently compute the correlation

functions from the trial microstructure. For the alloy micro-

structure, we incorporate the directional two-point correla-

tion function S2 and two-point cluster function C2 into the

reconstructions. The directional lineal-path function L is

employed to ascertain the accuracy of the reconstructions.

Since an extremely large number of trial microstructures are

generated, it is highly undesirable to completely re-compute

the functions for each trial. Instead, one can use the correla-

tion functions of the old microstructure to obtain those for

the new microstructure, which we discuss in detail in this

section.

In the reconstructions, we only consider S2 and C2 along

three orthogonal directions. Accordingly, the original 3D

microstructural array can be decomposed into three subsets

of 1D arrays. The 1D arrays in each subset are along one of

the three orthogonal directions. The correlation functions of

interest associated with the 1D arrays in a subset are com-

puted and averaged to obtain the corresponding directional

correlation function. Specifically, we consider the voxels in a

1D array as “particles” of a lattice-gas system. For the auto-

correlation functions, the voxels of the phase of interest are

grouped and the separation distance between a pair of voxels

within the group is computed. These distances are then

binned and normalized by the size of the 1D array to obtain

the corresponding correlation function. To obtain the

cross-correlation functions, the voxels of two different

phases are grouped and the separation distance between a

pair of voxels, one from each group, is computed, binned,

and normalized. For the cluster functions, the voxels belong-

ing to the same cluster of the phase of interest are grouped.

The pair separation distances within each group are com-

puted, binned, and normalized to obtain the directional C2.

In the generation of a new trial microstructure, the posi-

tions of a pair of randomly selected voxels belonging to

different phases are switched. For S2, only six 1D micro-

structural arrays, two from each subset for a specific direc-

tion, are affected. Accordingly, only the directional

correlation functions associated with these six 1D micro-

structural arrays need to be re-computed in order to obtain

the directional two-point correlation functions of the entire

new trial microstructure. This enables one to efficiently

re-compute S2 for the trial microstructures. For C2, one needs

to track the evolution of the clusters due to the voxel switch-

ing. There are three possible “cluster events” associated with

the voxel switching: (i) creation of a new cluster of a phase

(i.e., an isolated voxel surrounded by voxels of other phases);

(ii) combination of small clusters into a single cluster; (iii)

breaking of a cluster into small clusters. After each switch-

ing, possible cluster events are checked and the cluster con-

figuration in the microstructure is updated accordingly. The

directional C2 are then re-computed based on the updated

cluster configuration. Note that in general more than six 1D

arrays will be affected by the cluster events. Therefore,

reconstructions incorporating C2 are more time-consuming

than the S2-alone reconstructions.

IV. RESULTS

In the section, we will present our framework for model-

ing anisotropic heterogeneous materials via the directional

two-point correlation function S2 and directional two-point

cluster function C2, which is motivated by a number of stud-

ies showing that using separate structural information along

three orthogonal directions can lead to accurate reconstruc-

tions of anisotropic microstructures.29,43–46 Specifically, the

utility of the framework will be illustrated by applying it to

model an anisotropic alloy microstructure, which was

obtained via high-resolution x-ray tomographic microscopy

(Fig. 2). Aluminum (Al) alloys have been widely used in

many engineering structures especially in automotive and air-

craft applications due to their unique high strength-to-weight

ratio and corrosion resistance. An aluminum alloy almost

always has secondary phase inclusions and particles with

impurities that are present in the microstructure. Due to roll-

ing of the alloy, the inclusions usually process anisotropic

shapes, with the dimension along the rolling direction signifi-

cantly elongated. This in turn results in an overall anisotropic

alloy microstructure.

By quantitatively comparing various reconstructed

microstructures from different correlation functions with the

actual microstructure, we show that incorporating clustering

information can further improve the accuracy of the

FIG. 2. 3D alloy microstructure with anisotropic inclusions of Fe-rich

(shown in green) and Si-rich (shown in red) phases in Al matrix (transpar-

ent) obtained from x-ray tomographic microscopy. The size of the sample is

roughly 360 lm� 680 lm� 860 lm. This figure is reproduced by permis-

sion from Singh et al., Metall. Mater. Trans. A 43, 4470–4474 (2012).

Copyright 2012 by Springer Science & Business Media.

093511-4 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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reconstruction, although S2-alone reconstruction already

capture the salient morphological features of the alloy micro-

structure. In addition, we introduce a set of analytical “basis”

functions associated with elongated particles to approximate

the S2 and C2 of the inclusions. This allows us to characterize

the alloy with only six effective linear sizes of the Fe-rich

and Si-rich inclusions along three orthogonal directions.

A. Microstructure modeling

Figure 2 shows the 3D microstructure of the alloy

obtained from image segmentation and reconstruction of the

x-ray tomography data set. It can be clearly seen that both

the Fe-rich and Si-rich inclusions are elongated along the

RD. On the other hand, the inclusions along the TD and

ND directions are more isotropic. These result in an overall

anisotropic microstructure. Details of the x-ray tomography

process, conducted at the Advanced Photon Source, Argonne

National Laboratory, are reported elsewhere.47

Figure 3 shows the directional two-point correlation func-

tion S2 and two-point cluster function C2 associated with the

inclusion phases. The functions associated with the actual

microstructure are sampled from the complete 3D microstruc-

tural array in order to correctly capture the clustering informa-

tion. It can be seen that the autocorrelation functions SFe2 and

SSi2 almost monotonically decay to the associated long-range

asymptotic values ui2. Moreover, the cross-correlation func-

tion SFe�Si2 rapidly increases to the associated asymptotic

value uiuj. These behaviors suggest that inclusions almost

have no spatial correlations between one another, i.e., they are

nearly randomly distributed in the microstructure. This is

because the volume fractions of the inclusions are extremely

small, and thus, the system can be considered at the low ulimit. As we will quantitatively show below, at this limit, the

microstructure is mainly determined by the distribution of the

shape and size of the inclusions. The two-point cluster func-

tion C2 associated with the inclusion phases monotonically

decay to zero. Similar to the autocorrelation functions, the

range of C2 along the RD is much longer than the other two

directions, clearly indicating that the clusters (inclusions) are

elongated along this direction.

Here, we approximate the autocorrelation function of

the inclusion phase a with the following function:

Sa;b2 ðrÞ ¼ ua Aa

b expðBab � rÞ þ ð1�Aa

bÞVintðr; �DabÞ

h inþu2

a 1�Vintðr; �DabÞ

h ioHð �D

ab � rÞ þu2

aHðr� �DabÞ;

(8)

where ua is the volume fraction, Aab, Ba

b, and �Dab are parame-

ters to be determined. The approximation (9) includes two

“basis” functions: the Debye random medium function

FIG. 3. Directional correlation functions associated with the inclusion phases in the Al alloy. (A) Directional two-point correlation functions SFe2 of the Fe-rich

inclusions. (B) Directional two-point correlation functions SSi2 of the Si-rich inclusions. (C) Directional cross-correlation functions SFe�Si

2 associated with the

inclusion phases. (D) Directional two-point cluster functions CFe2 of the Fe-rich inclusions. (E) Directional two-point cluster function CSi

2 of the Si-rich inclu-

sions. The unit of length in the plots is the linear size of a cubic voxel, which is �1.8 lm.

093511-5 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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exp(Br)26,48 and Vint(r; D),1 which is the scaled intersection

volume of two isotropic inclusions with linear size D sepa-

rated by r:

Vintðr; DÞ ¼ 1� 3r

2Dþ r3

2D3(9)

and HðxÞ is the Heaviside function, i.e.,

HðxÞ ¼1 x � 0

0 x < 0:

((10)

In other words, one can consider the alloy microstructure as

a mixture of “clusters of all shape sizes” (features of a

Debye random medium) and well defined inclusions (charac-

terized by Vint). For the anisotropic inclusions, we consider

that each direction possesses distinct combination parameter

Aab, correlation parameter Ba

b, and effective size �Dab, which is

the effective linear size or “length” of the inclusions along

that direction. The values of these parameters are determined

such that the approximated functions best match the actual

autocorrelation function (i.e., the squared difference between

two functions is minimized) and the value of the effective

inclusion lengths are obtained via this fitting procedure. In

Tables I and II, we provide the values of the parameters for

the Fe-rich and Si-rich inclusions in the alloy, respectively.

Note that we do not model the cross-correlation function

SFe�Si2 since at the low u limit its values are several orders of

magnitude smaller than the autocorrelation functions, which

again suggests that there are no significant inter-particle spa-

tial correlations.

By definition, the two-point cluster function C2 meas-

ures the contributions from the point pairs in the same cluster

(inclusion) in S2; see Eq. (5). Thus, we approximate C2 of

the inclusion phase a as follows:

Ca;b2 ðrÞ ¼ua Aa

b expðBab � rÞ þ ð1� Aa

bÞ 1� 3r

2 �Dab

þ r3

2 �Dab

!" #

�Hð �Dab � rÞ; ð11Þ

where the parameters Aab, Ba

b, and �Dab are given in Tables I

and II, respectively, for the Fe-rich and Si-rich inclusions.

We note that these parameters are identical for S2 and C2,

because C2 is the connectedness portion of S2. Thus, the

approximated C2 does not provide additional mathematically

information concerning the material microstructure besides

that contained in corresponding S2. However, as we will

show in the next subsection, incorporating C2 in the recon-

struction procedure to constrain clustering in the

microstructure will improve the accuracy of the reconstruc-

tions. In addition, we emphasize that although the S2 and C2

appear to be close to one another in Fig. 3 (due to the small

inclusion volume fractions), these are intrinsically different

quantities in that S2 is long-ranged and C2 decays to zero

beyond the largest linear inclusion size.

We note that in Eqs. (8) and (11), a number of “basis”

functions have been employed to approximate the correlation

functions of the material. It has been shown26,36 that the

collection of all realizable standard two-point correlation

functions S2 span a well-defined correlation function space.

This implies that in principle, the standard two-point correla-

tion function of an arbitrary heterogeneous material can be

exactly represented as the linear combination of a complete

set of basis functions. In practice, such a complete set of

basis functions are extremely difficult to obtain (if not

impossible) and thus, an incomplete subset of basis functions

(including the ones used here) have been suggested,26 which

are associated with well known model microstructures with

distinct salient morphological features. Although empirically

chosen, the subset of basis functions has been shown to be

able to represent a wide spectrum of heterogeneous material

microstructures26,27 and allows one to “decompose” the

structural features of the material of interested into a set of

key features associated with very known model structures.

The space of two-point cluster functions C2 has not been

systematically investigated before. However, as indicated in

Sec. II C, C2 is the connectedness contribution to the standard

two-point correlation function S2. Therefore, it can be

expected that the connectedness portions of the basis functions

for the S2 space provide a natural choice of the basis functions

for the C2 space. Here, we have used the intersection-volume

function Vint(r) as one of the basis functions to approximate C2

of the inclusion phases, which is the connected contribution to

the S2 of a binary material with dilute distribution of

non-overlapping particles in a matrix. A rigorous mathemati-

cal treatment of the correlation function space is out of scope

of the current paper and interested readers are referred to Refs.

26 and 36 and references therein for detailed discussions.

B. Microstructure reconstruction

We employ the stochastic reconstruction procedure

discussed in Sec. III to determine the minimal set of structure

information required to statistically characterize the aniso-

tropic alloy microstructure. Specifically, we consider the

reconstructions from the directional S2 alone and from the

combination of directional S2 and C2, including those

obtained from the experimental data and the analytical

approximation. We note that only autocorrelation functions

TABLE I. The values of Aab, Ba

b, and �Dab for the Fe-rich inclusions along

three orthogonal directions in the alloy.

Fe-rich inclusions RD TD ND

Aab 0.801 0.850 0.902

Bab �0.350 �0.504 �0.579

�Dab 81.2 32.0 18.9

TABLE II. The values of Aab, Ba

b, and �Dab for the Si-rich inclusions along

three orthogonal directions in the alloy.

Si-rich inclusions RD TD ND

Aab 0.918 0.898 0.951

Bab �0.480 �0.548 �0.679

�Dab 86.8 25.9 11.0

093511-6 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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of the inclusions are used for the reconstructions. Figure 4

shows the visual comparison of a portion of the actual alloy

microstructure with various reconstructions. The microstruc-

tures are visually indistinguishable from one another.

Figure 5 shows the lineal-path functions L of the Fe-rich

and Si-rich inclusions in the actual and reconstructed micro-

structures. This quantity is employed to ascertain the accuracy

of the reconstructions because it contains lineal connectedness

information of the inclusion phases, and thus, is sensitive the

degree of clustering in the system. It can be seen that L associ-

ated with different microstructures lie closely to one another,

which is consistent with the visual comparison. However,

quantitative differences can still be observed. The lineal-path

functions of the S2-alone reconstruction are slightly longer

ranged than those of the actual microstructure, indicating

slight overestimation of the degree of clustering in the recon-

struction, which is consistent with the observation report in

Ref. 29. On the other hand, reconstructions incorporating C2

can reproduce the degree of clustering very well, as measured

by L. This is not surprising since C2 contains the complete

connectedness information while L only contains partial infor-

mation, as we discussed in Sec. II.

Importantly, one can see that the reconstruction from the

approximated S2 and C2 can very well reproduce the salient

structural features of the alloy. This suggests that the alloy

microstructure can be statistically characterized by the pro-

posed analytical approximations of the correlation functions,

which depend only on parameters given in Tables I and II. In

other words, with our choice of the basis functions, the 3D

microstructural array specifying the local state of each indi-

vidual voxels can be reduced to a handful of scalar parame-

ters that statistically model the alloy microstructure.

V. SUMMARY AND CONCLUDING COMMENTS

We have provided a framework to model the microstruc-

ture of a heterogeneous material with anisotropic inclusions

secondary phases via directional correlation functions of the

inclusions and stochastic reconstructions. Specifically, we

proposed a set of “basis” functions parameterized by the

effective linear size of the inclusions to approximate the

directional two-point correlation function S2 and two-point

cluster function C2 of the inclusion phases. By devising an

efficient method that enables us to incorporate directional C2

into the reconstruction procedure, we quantitatively showed

that the analytical approximations can lead to reconstructions

that very well reproduce the salient structural features of the

material. This suggests that the heterogeneous microstructure

can be statistically modeled by a handful of parameters,

which significantly reduces the set of parameters required to

characterize the material.

In general, directional S2 alone is not sufficient to char-

acterize a microstructure even in a statistical sense.49,50 It

has been well established that S2-alone reconstructions

FIG. 4. Visual comparison of the actual alloy microstructure with various reconstructions. The Fe-rich inclusions are shown in blue and Si-rich inclusions in

red. The Al matrix is transparent. The size of samples is 102� 192� 248 voxels (�180 lm� 340 lm� 430 lm). (A) A portion of the actual alloy microstruc-

ture obtained from tomography. (B) S2 alone reconstruction. (C) S2-C2 reconstruction from the correlation functions associated with the actual microstructure.

(D) S2-C2 reconstruction from the approximated correlation functions.

FIG. 5. Quantitative comparison of the

directional lineal-path functions of the

actual alloy microstructure and various

reconstructions. (A) Fe-rich inclusions.

(B) Si-rich inclusions. The unit of

length in the plots is the linear size of a

cubic voxel, which is �1.8 lm.

093511-7 Y. Jiao and N. Chawla J. Appl. Phys. 115, 093511 (2014)

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usually significantly overestimate the degree of clustering in

the system.30 Thus, the alloy microstructure studied here is a

special case where S2 alone can lead to fairly accurate recon-

structions. This is because the system is in the low

volume-fraction limit. During the reconstruction process, it

is very difficult to form large clusters due to the small num-

ber of voxels associated with the inclusion phases.

Nonetheless, the reconstruction can be further improved by

incorporating the cluster function C2. At the low u limit,

there are not significant spatial correlations between the

inclusions, and thus, the cross-correlation function becomes

irrelevant to the reconstruction. For systems at higher vol-

ume fraction, inter-phase correlations become crucial to

determining the microstructure and physics-based models

(e.g., hard-particle packings51) should be employed to deter-

mine and approximate such correlations.

The general framework to model complex microstruc-

tures via correlation functions can be applied to other hetero-

geneous materials such as composites and porous media as

well. Specifically, the two-point correlation function space

alone has been shown to be useful for the modeling of a vari-

ety of complex microstructures.26,27 It can be expected that

the incorporation of additional correlation functions can lead

to more accurate characterization and modeling of heteroge-

neous material microstructure, as we explicitly demonstrated

in this paper. In general, this framework allows one to quan-

titatively characterize and control material microstructure

via simple analytical functions. This will allow one to

parameterize the correlation functions with material process-

ing conditions (such as annealing time30), and thus establish

the quantitative processing-microstructure relation.

Structure-based analysis of the properties and performance

of the material can be carried out to establish rigorous

processing-structure-property relations. This would lead to

the development of novel material design procedure, which

we will explore in future studies. Finally, 3D material micro-

structure modeling via correlation functions is also common

in small angle x-ray scattering52 and grey-tone electron

micrograph analysis.53 It would be interesting to apply the

analytical representation scheme for correlation functions in

these situations.

ACKNOWLEDGMENTS

This work was supported by the Division of Materials

Research at National Science Foundation under Award No.

DMR-1305119 and the Office of Naval Research (ONR)

under Contract No. N00014-10-1-0350 (Dr. A. K.

Vasudevan, Program Manager). Use of the Advanced Photon

Source was supported by the U.S. Department of Energy,

Office of Science, Office of Basic Energy Sciences, under

Contract No. DE-AC02-06CH11357. The experimental as-

sistance of Jason Williams and Sudhanshu Singh at ASU,

and Xianghui Xiao and Francesco De Carlo at APS are grate-

fully acknowledged.

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